$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Verification: quadratic solution (1)
Manufactured solution:
$$
\begin{equation}
\uex(x,y,t) = x(L_x-x)y(L_y-y)(1+{\half}t)
\tag{43}
\end{equation}
$$
Requires \( f=2c^2(1+{\half}t)(y(L_y-y) +
x(L_x-x)) \).
This \( \uex \) is ideal because it also solves the discrete equations!
